Title: Multivariate Wold decompositions
Authors: Federico Severino - Universite' Laval (Canada) [presenting]
Fulvio Ortu - Universita Bocconi (Italy)
Claudio Tebaldi - Universita Bocconi (Italy)
Simone Cerreia-Vioglio - Universita Bocconi (Italy)
Abstract: Multivariate time series are driven by a collection of (possibly correlated) univariate shocks that need to be properly identified in the applications. Moreover, in many economic contexts, the process under scrutiny is the outcome of the superposition of simultaneous disturbances with heterogeneous frequencies that can generate short-, medium- or long-term effects. Given a weakly stationary vector process, we provide a methodology to elicit uncorrelated persistent components driven by multivariate shocks with increasing duration: the Multivariate Extended Wold Decomposition. By introducing multivariate scale-specific responses, we can quantify the persistence in vector autoregressive models, once their shocks are identified. To derive the decomposition, we embed the vector process in a Hilbert A-module framework where matrices replace the field of scalars, and we prove the Abstract Wold Theorem for self-dual pre-Hilbert A-modules. From this abstract result, by using projection techniques, we retrieve the well-known Multivariate Classical Wold Decomposition. We also derive the persistence-based Multivariate Extended Wold Decomposition. We finally apply the latter to some well-known macroeconomic bivariate VAR models.